We determine, up to lower-order terms in the exponent, the best possible deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix. If $A\succeq 0$ has no zero diagonal entry, $d=\operatorname{rank}(A)$, $A=VV^\dagger$ with $V\in\mathbb{C}^{n\times d}$ full column rank, and $v_1,\ldots,v_n$ are the rows of $V$, define \[ Φ(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{Φ(V)}. \] We prove the exact sandwich \[ e^{-γn}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). \] Here $γ$ is the Euler--Mascheroni constant. Since the maximization is concave, this gives a deterministic polynomial-time $e^{(γ+\varepsilon)n}$-approximation for every $\varepsilon>0$. Combined with the previous $e^{(γ-\varepsilon)n}$-hardness of approximation for positive semidefinite permanents, this resolves the optimal exponential approximation ratio for deterministic polynomial-time algorithms as $e^{(γ+o(1))n}$, assuming $\mathrm{P}\ne\mathrm{NP}$. The proof is an entropy argument applied to the standard Wick integral formula for $\operatorname{per}(A)$; the loss is exactly $γ$ per factor because $\mathbb{E}[\log T]=-γ$ for $T\sim\operatorname{Exp}(1)$. The result was obtained through interactions with GPT 5.5 Pro Extended: the first author's interaction was one-shot, while the second author's was a separate multi-turn interaction with high-level guidance. Both authors verified the theorem and proof. Codex was used to assemble and typeset the manuscript.

翻译：我们确定了厄米正半定矩阵积和式的最佳确定性多项式时间近似比（精确到指数低阶项）。若 $A\succeq 0$ 无零对角元，$d=\operatorname{rank}(A)$，$A=VV^\dagger$ 且 $V\in\mathbb{C}^{n\times d}$ 列满秩，令 $v_1,\ldots,v_n$ 为 $V$ 的行向量，定义 \[ Φ(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{Φ(V)}. \] 我们证明精确的夹逼关系 \[ e^{-γn}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). \] 其中 $γ$ 为欧拉-马歇罗尼常数。由于最大化问题是凹的，这给出了对所有 $\varepsilon>0$ 的确定性多项式时间 $e^{(γ+\varepsilon)n}$-近似。结合先前对正半定积和式的 $e^{(γ-\varepsilon)n}$-困难性结果，这确定了在 $\mathrm{P}\ne\mathrm{NP}$ 假设下确定性多项式时间算法的最优指数近似比为 $e^{(γ+o(1))n}$。证明是对 $\operatorname{per}(A)$ 的标准Wick积分公式运用熵论证；每个因子损失恰好为 $γ$，因为当 $T\sim\operatorname{Exp}(1)$ 时 $\mathbb{E}[\log T]=-γ$。该结果通过与GPT 5.5 Pro Extended的交互获得：第一作者的交互为单次完成，而第二作者为另一组多次交互并接受高水平指导。两位作者验证了定理及证明。使用Codex完成手稿的汇编与排版。